The admission fee at an amusement park is $2.00 for children and $6.80 for adults.

Question

The admission fee at an amusement park is $2.00 for children and $6.80 for adults. On a certain day, 285 people entered the park, and the admission fees collected totaled 1458 dollars. How many children and how many adults were admitted?

number of children equals
number of adults equals

Answer 1

The way you will set this question up is key. This will be a substitution formula and there will be two equations.

Let C equal the number of children and A equal the number of adults. 

The two equations are:

A + C = 285 {the number of adults + the number of children = the number of people}

A(6.80) + C(2.00) = 1458 {the number of adults times the price of adults + the number of children times the price of children = the total admission fees}

I think it's easiest to rearrange the first equation and substitute it into the second equation. Subtract C from both sides of the first equation and you have A = 285 - C. Substitute this new sentence into the second one, since there will only be one term in it.

(285 - C)(6.80) + C(2.00) = 1458 {we put the first equation in for A}

1938 - C(6.80) + C(2.00) = 1458 {we multiplied (285 -C) by 6.80}

1938 - C(4.80) = 1458 {we simplified the C term}

- C(4.80) = -480 {we subtracted 1938 from both sides}

C(4.80) = 480 {multiplied both sides by -1}

C = 100 {divided both sides by 4.80)

Now we take our answer for C and go back to our first equation to solve for A

A + C = 285

A + 100 + 285 {put in value of C}

A = 185 {subtracted 100 from both sides}

Number of children = 100 & the number of adults = 185

You can check your work by plugging your values for A & C into the second equation

185(6.80) + 100(2.00) = 1458

1258 + 200 = 1458

Correct! :)

Answer 2

You are given the following:  

     Admission fee for children = $2.00

     Admission fee for adults = $6.80

     Total admission fees collected = $1,458

     Total # of people admitted (children + adults) = 285 

Let:     x = # of children admitted

          y = # of adults admitted

With the given information, we can generate a system of linear equations one of which will yield the total # of people admitted into the park and the other will yield the total admission fees collected.

Since x is the # of children and y is the # of adults, and we know that the total # of people is 285, then we arrive at the following:     x + y = 285

Since the admission fee for children is $2.00 per child then the fee for children multiplied by the # of children (x) gives you the fees collected for the children admitted. Similarly, the fee for adults ($6.80 per adult) multiplied by the # of adults (y) gives you the fees collected for the adults admitted. Adding the fees collected for the children to the fees collected for the adults yields the total admission fees collected ($1,458). That is,

     (2.00)x + (6.80)y = 1,458

With this, the system consists of the following equations:

          x  +     y  =  285

        2x  + 6.8y = 1,458

There are a couple of ways to solve for the system, I find the simplest to be the elimination method. Using this method, we eliminate one of the variables by multiplying one of the equations by a constant that will generate a coefficient for the this variable that is the opposite of the same variable's coefficient in the other equation. For instance, if we choose to eliminate the x variable then we multiply the first equation by -2 to yield -2x which will be eliminated since the second equation has a +2x:

     -2(x + y = 285)     ==>     -2x - 2y = -570

Now we add this equation to the second equation:

       -2x  -    2y = -570

+      2x + 6.8y = 1,458

_______________________

        0x + 4.8y = 888       ==>       4.8y = 888

Solve for y by dividing both sides of the equation by 4.8:

     4.8y/4.8 = 888/4.8

               y = 185

Use the answer for y to solve for x by plugging in its value in the first original equation:

     x + y = 285

     x + 185 = 285

Subtract 185 from both sides of the equation to solve for x:

     x + 185 - 185 = 285 - 185

     x = 100

Solution:     x = 100     and      y = 185

Thus, the number of children admitted is 100 and the number of adults admitted is 185.

Answer 3

Okay, here's how you figure this out:  you use what they call systems of equations.  We know that the the number of children plus the number of adults is 285.  After all, EVERYBODY is either an adult or a child.  We also know that 2 times the number of children plus 6.8 times the number of adults adds up to 1458. 

Now here's how we use the system.  First, let's assign a variable to children and to adults.  Let's say the number of adults is X and the number of children, Y.  In that case, we know the following:

X + Y = 285     AND WE KNOW:    6.8x + 2y = 1458

Now what we will do is solve for one of these variables.  Either X or Y will do.  I decided to solve for X.  Therefore, what I will do is use the substitution method to get X by itself.  How do I do that?  BY FINDING A WAY TO EXPRESS Y IN TERMS OF X. 

Since X + Y = 285 in one of the equations, I can take this equation, subtract X from each side, and then get Y = 285 - X  

Now that I have solved for Y in the first equation, I can SUBSTITUTE that value of Y in for "Y" in the second equation and BINGO, I have an equation entirely written in terms of X.

6.8x + 2y = 1458 will now turn into....    6.8x + 2(285 - X) = 1458

See what I just did?  I plugged the (285 - X) value of Y in for the Y value of the second equation.  Now I can distribute:  6.8x + 570 - 2x = 1458 is the new value of the equation after I do this.  Then I combine like terms on the left side:

4.8x + 570 = 1458   Now I'll subtract 570 from each side:

4.8x = 888  And now I divide each side by 4.8:

X = 185.

So there you go.  The value of X, the number of adults, is 185.  Since there were 285 people, the number of children must therefore have been 100. 

For best results, make sure your values work in both equations:

X + Y = 285      Now plug and chug to get 185 + 100 = 285.  A true statement

6.8x + 2y = 1458  Now plug and chug to get 6.8(185) + 2(100) = 1458.  Another true statement.  

There were, therefore, 185 adults and 100 children.